We could use copy-on-write approach instead of copying on every pulse;

This means, in synchronized block, we'd set some flag:
synchronized(this) {
   stagePulseListInUse = scenePulseListInUse = postScenePulseListInUse = true;
}

and unset them at the end of the pulse.

Then instead of full copies, do just reference copy:
final Map<TKPulseListener,AccessControlContext> stagePulseList = stagePulseListeners;
...

The add*Listener methods would look like:
 
synchronized(this) {
    AccessControlContext acc = AccessController.getContext();
    if (stagePulseListInUse) {
        stagePulseListInUse = false;
        stagePulseListeners = new WeakHashMap<>(stagePulseListeners);
    }
    stagePulseListeners.add(listener);
}